Robust Time Series Analysis
Robust Time Series Analysis (M- and MM-estimation based AR / MA / ARIMA) · Also known as: robust ARIMA, robust autoregressive model, outlier-resistant time series, Robust Zaman Serisi Analizi
Robust Time Series Analysis fits autoregressive, moving-average, and ARIMA models to series that contain outliers or structural breaks, using M-estimation or MM-estimation instead of ordinary least squares so that a few anomalous observations do not distort the fit. It follows the robust statistics tradition consolidated in Maronna, Martin, Yohai and Salibián-Barrera (2019).
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When to use it
Use robust time series analysis when you have a single continuous series over time (at least about 50 observations) for forecasting or prediction, and you suspect the data contain outliers, anomalous spikes, or structural irregularities that would corrupt a classical AR / ARIMA fit. It does not require normally distributed errors. It is unsuitable for very short series — below roughly 20 points the robust AR / ARIMA parameters cannot be estimated reliably — and if more than about a quarter of the observations are contaminated, an M-estimation model breaks down and a median-based filter is preferable.
Strengths & limitations
- Resists outliers and structural breaks: a few anomalous observations do not derail the estimated dynamics.
- Does not assume normally distributed errors, so it handles heavy-tailed innovations.
- MM-estimation combines a high breakdown point with high efficiency, giving reliable parameters on contaminated data without sacrificing much precision on clean data.
- Needs a reasonable series length (about 50 observations); below roughly 20 points the robust AR / ARIMA parameters cannot be estimated reliably.
- When the outlier ratio exceeds about 0.25, M-estimation itself breaks down and a median-based filter is preferable.
- More computationally involved and harder to interpret than a classical least-squares time series model.
Frequently asked
How is this different from a standard ARIMA model?
The model equation is the same AR / MA / ARIMA structure, but the parameters are fitted by minimising a bounded loss (M- or MM-estimation) rather than the sum of squared residuals. This caps the influence of outliers and structural breaks, so a few anomalies cannot dominate the estimated dynamics.
What is the difference between M-estimation and MM-estimation here?
M-estimation down-weights large residuals through a bounded loss function. MM-estimation goes further by first securing a high breakdown point and then refining for efficiency, so it can tolerate a substantial fraction of contamination while staying nearly as precise as least squares on clean data.
How many observations do I need?
About 50 is a sensible minimum. Below roughly 20 observations the robust AR / ARIMA parameters cannot be estimated reliably, and a permutation-based approach is more appropriate.
What if my series is heavily contaminated?
When more than about a quarter of the observations are outliers, M-estimation based time series models break down. In that regime a median-based filter, such as a MAD-based estimator, is the safer choice.
Sources
- Maronna, R. A., Martin, R. D., Yohai, V. J., & Salibián-Barrera, M. (2019). Robust Statistics: Theory and Methods (with R) (2nd ed.). Wiley. ISBN: 978-1119214687
- Peña, D., & Guttman, I. (1988). A Bayesian Approach for Predicting with Outliers. Journal of the American Statistical Association. link ↗
How to cite this page
ScholarGate. (2026, June 1). Robust Time Series Analysis (M- and MM-estimation based AR / MA / ARIMA). ScholarGate. https://scholargate.app/en/statistics/robust-time-series
Which method?
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